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Symbolism:
70 corresponds to the totality of an evolution, an evolutionary
cycle being fully completed, according to Saint Augustin.
The corresponding Hebraic letter is 'ain, which corresponded
to the sixteenth mystery of the Tarot: the house-God, symbol of
fall, collapse, catastrophe - there is agreement with the idea
of termination of cycle, according to R. Allendy.
In the Bible:
The 70 weeks prophecy of the book of Daniel. (Dn 9,24)
The 70 years of captivity of Babylonian of two of the twelve tributes
of Israel. (2 Ch 36,21)
Ezekiel sees 70 ancients adoring the idols. (Ezk 8,11)

is
the smallest number that can be written as a sum of two primes,
each ending with the digit 1, in two distinct ways:
72 = 11 + 61 = 31 + 41
is the largest number of spheres that can touch another sphere in
a lattice packing in 6 dimensions.
Dividing 72 by the interest rate (x percent),
you obtain the number of years you should wait to double
your initial capital.

is
a sexy
prime and an emirp.
is the smallest prime whose digits are reversed
in base 22 and the smallest number (besides 1) which is one less
than the double of its reverse: 2 x 37 - 73 = 1
is the first difficult prime to form under the rules of the four
4's puzzle (The goal of the puzzle is to find the simplest
mathematical expression for every integer, using only common mathematical
symbols and the digit four).
is a divisor of 104 + 1 = 10001 (composite
Fermat number).

Every
positive integer can be expressed as a sum of 73 or fewer sixth
powers.

The
mirror of 73 - the 21st prime number - is 37 - the 12th prime
number.

(each
number of the sum represents a letter in its alphabetical
order)

73 =
1118

73
= (9! + 7! ) / 7!

is
the smallest number with the property that it and its neighbors
can be written as a sum of 2 squares.
72 = 62 + 6273 = 32 + 82
74 = 52 + 72- Found by Daniel Clemente from
the Polytechnical University of Catalonia

Most
buses in fiction novels display the route number 73. Actually,
many writers and authors chose 73 as the "archetypal" bus
route.

Any
number having a abcabc numeral
pattern is divisible by 77 (replace each letter with a digit.
E.g. 123123)

Seventy-seven
is the smallest number requiring five syllables in English.

French
speaking people have two different names for seventy-seven: 'Soixante-dix-sept'
(sixty and seventeen) and 'septante-sept' (Swiss French
people).

During
World War II in Sweden at the border with Norway, "77" (sjuttiosju)
was used as a 'Shibboleth' (discriminatory password), because
the tricky pronunciation in Swedish (approx. 'shütioohshü')
made it easy to instantly discern whether the speaker was native
Swedish, Norwegian, or German.

Renovation
of a landmark Los Angeles restaurant has revealed a neon lamp
that has been left on for about 77 years! Source: Dailymail.

Conspiracy
theorists claim that the number 77 appears
statistically in many mass death, e.g., Flight 77 crashing
into the Pentagon on September 11, or the infamous 7 July 2005
London bombings, also known as 7/7.

78

is
the lowest number which can be written as the sum of of 4 distinct squares in
3 ways.
is a triangular number (sum of consecutive integers
from 1 to 12):
78 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12

= 37
+ 41 (sum of consecutive primes)

783 =
393 + 523 + 653

Front
loaded interest is often offered on automobile loans. Front loaded
interest is also referred to as the Rules of 78 because
your total interest costs for the loan are divided into "78
parts" (sum of the digits of 12 months of a year). Those
78 parts are then paid during the first 12 months of the loan
according the following equation:
12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 78
The first month, you pay 12 of the 78 parts of the interest;
The second month, you pay 11 of the 78 parts;
The third month, you pay 10 of the 78 parts;
... and so on until the twelfth payment of 1 part interest.

The 78 RPM
phonograph records were the standard in 1926.
The common name for such records as 78s (or "seventy-eights")
as they played at 78 revolutions per minute.

79

is
a permutable prime,
and the smallest prime whose sum of digits is a 4th power.
is a Pillai
prime. A Pillai prime is a prime p for which there
is an integer n > 0 such that the factorial of n is
1 less than a multiple of the prime, but the prime is not 1 more
than a multiple of n, in short: n! ≡ -1
(mod p) but p ≠ 1 (mod n)

79 =
11 + 31 + 37 and
97 = 11 + 13 + 73 (sums of reversible primes)

=
(7 x 9) + (7 + 9) (this also works for 19, 29,
39, 49, ...)

=
402 - 392 = 40 + 39

= 27 - 72
= 22 + 52 + 52 + 52 (sum
of squares)

≈ 8.8882

x2 - 79x
+ 1601, if you replace x in the previous expression with any
number from 1 to 79, you always obtain a prime number!

is the
smallest number that is diminished by summing the alphabetical
order of its letters (writing out its English name and adding
the letters using a = 1, b = 2, c = 3, etc.): EIGHTY = 5 + 9
+ 7 + 8 + 20 + 25 = 74.

The "80:20
rule" originated from Vilfredo Pareto, an Italian economist.
He discovered a common phenomenon: about 80% of the wealth in
most countries was controlled by a consistent minority -- about
20% of the people.

In French
there are 3 ways to say 80:
'quatre-vingts', in France, and Quebec,
'huitante', in Switzerland,
and 'octante', in Belgium.

Spring
begins on March 21st, the 80th day of the year.

1 mile
= 8 furlongs = 80 chains

According
to U.S. statistics, an American woman would kiss an average of 80
men before she gets married.

"Around
the World in Eighty Days" (French: Le
tour du monde en quatre-vingts jours) is a classic adventure
novel by the French writer Jules Verne.

is
a Heptagonal
number, that is a polygonal number of the form n(5n -
3)/2
is the square of
the sum of its digits; there are no other numbers with this property
except 0 and 1.
is the 3rd lowest prime power of a prime power of a prime.

x2 - 81x
+ 1681, if you replace x in the previous expression with any
number from 1 to 80, you always obtain a prime number!

The
reciprocal of 81 gives an interesting recurring
decimal expansion:
1/81 = 0.123456790123456790123456790...
The result is actually a repeating decimal containing every digit
from 0 to 9, except for 8. - Posted by Ting WangHowever, the smallest fraction that gives a repeating
set of all the digits 0-9 placed successively is:
13717421/111111111
= 0.1234567890123456789...

is
an happy
number, and a companion Pell
number.
is a 15-gonal number and a centered 27-gonal number.
is the number of different ways you can arrange 6 regular hexagons
by joining their adjacent sides; thus it is the number of possible 6-hexes (with
82 6-hexes it is possible to form an hexagonal
ring).
= 1 + 5 + 21 + 55 (sum of Fibonacci
numbers)
= 19 + 20 + 21 + 22 (sum of consecutive integers)
= 12 + 92= 22 + 22 + 52 + 72 (sum
of square primes)= 2(42 + 52)

is
one of the nuclear
magic numbers in physics. 'Nuclear magic numbers' are
the numbers of nucleons (protons or neutrons) forming full outer
shells in an atomic nucleus, that are relevant for nuclear isotope
stability.
is the international telephone dialing code for Korea.
is the International Standard Book Number for books published in
Norway (ISBN 82-...).
The verb to pay is used 82 times in the Bible.

The Latin
and Greek prefixes "duooctoginta-", "dioctaconta-", "dooctaconta-", "octacontadi-",
and/or "octacontakaidi-" all mean 'eighty-two'.
The noun and adjective "duooctogintennary" means 'a group
of 82' or 'a 82-year period'. The adjective and noun "duooctogintennial" means
'recurring every 82 years' or 'lasting 82 years'. The noun and
adjective "duooctogintennium" means 'a period of 82 years'.

The number
82 appears frequently throughout the film "Magnolia",
written and directed by Paul Thomas Anderson, and starring
Tom Cruise.

82nd
Verse in Tagore's Gitanjali:Time is endless in thy hands, my Lord. There
is none to count thy minutes.
Days and nights pass and ages bloom and fade like flowers. Thou
knowest how to wait.
Thy centuries follow each other perfecting a small wild flower.
We have no time to lose, and having no time we must scramble for
a chances. We are too poor to be late.
And thus it is that time goes by while I give it to every querulous
man who claims it, and thine altar is empty of all offerings to
the last.
At the end of the day I hasten in fear lest thy gate to be shut;
but I find that yet there is time.

83

is
the number of zeroless squares that are composed of the digits 1
up to 9. - Beilier

= 11
+ 31 + 41 (sum of the first 3 primes ending with
1)

it seems
there are over 1083 elementary particles with non-zero
mass in the known universe.

is
an hexadecagonal number and the only number known whose square has
no isolated digits. In
fact, its square has all its digits twice: 882 = 7744
is an 'untouchable
number'. An untouchable number is a positive integer that cannot
be expressed as the sum of all the proper divisors of any number.
The sequence of the first untouchable numbers is: 2, 5, 52, 88,
96, 120, 124, 146, 162, 188, 206, 210...

Eighty-eight,
11 letters long, is the longest number that is normally typed
using strictly alternating hands (ignoring the hyphen).

88 is
used in Chinese culture to mean "bye bye"; found in
Chinese-language chat, text, SMS, IM. 88 is
in fact pronounced in Chinese Mandarin language as "ba ba",
simulating the sound of the English language farewell "bye
bye".

The 8th letter
of the alphabet is 'H'. 8 two times signifies 'HH', shorthand
for the Nazi greeting:"Heil Hitler". 88 is
often found on hate group flyers, in both the greetings and closing
comments of letters written by neo-Nazis, and in e-mail addresses.

89

is
a prime and a Fibonacci number.
generates a cyclic number series where each term is the sum of
the square digits of the previous number:
89 - 145 - 42 - 20 - 4 - 16 - 37 - 58 - 89 - ...

The
decimal expansion of the inverse of
89 (that is 1/89) is just the Fibonacci series, added together
in an appropriate fashion: like a column of decimal fractions,
arranged so the right most digit of the nth Fibonacci
number is in the n+1th decimal place:

is the
product of 2 consecutive integers. Such specific numbers are
called 'oblong
numbers' because they correspond to the area of an "oblong" rectangle
with sides n and n+1. Thus, oblong numbers
can also be expressed as n2 + n.

is the
number of degrees in a right angle. Moreover, as a compass direction,
90 degrees corresponds to east.

Some
'scientific' reviews state humans are "90% water" when
in fact newborns have approx. 78%, 1-year-olds around 65%, adult
men about 60%, and adult women around 55%. See mythical
numbers.

"90-60-90" is
considered as an ideal proportion for becoming a model (90-60-90
are the breast-waist-hips measurements in centimeters, these
were the alleged measurements of Marilyn Monroe).

Bamboo
can increase its height by 90 centimeters in just one day.

The
Greek and Latin prefixes "enneaconta-", "nonaconta-",
and "nonaginta-" respectively means 'ninety'.
"Nonagerian" is a person who is from 90 to 99 years old.
The compound noun and adjective "fourscore and ten" means
'ninety'.

Even
if you eat a fully organic diet, you would still eat ninety E-numbers (in
fact, our body produces four E-numbers itself). E-numbers are
codes for chemicals used as food additives within the European
Union.

In
Italian, pezzo da novanta (piece of ninety) means 'big
boss' or 'big shot'. In Italy, when someone is or has been frightened
by something unexpected, it is used to say la paura fa novanta! (the
scare does ninety!).

is the
number of different arrangements of 8
non-attacking Queens on an 8 by 8 chessboard (i.e. no two
Queens should share the same row, column, or diagonal).

Multiply
92 by 8, then that product by 8, then that product by 8, and so on...
List the products one under the other, shifting the digits two places
to the right as shown below. Continue indefinitely. Add. The sum
converges to a string of 9s:

92
736
5888
47104
376832
3014656
24117248
192937984
9999999...

93

=
333 in base 5.
With just 9 straight cuts a potato can be divided into 93 pieces.

= 22 +
52 + 82(sum of squares)= 13 + 14 + 15 + 16 + 17 + 18

94

is
a 17-gonal number and a Smith
number (a Smith number is a composite number the sum of whose
digits is the sum of the digits of its prime factors excluding
1, here: 9 + 4 = 2 + 4 + 7).

94! -
1 is prime.

95

is
the number of planar
partitions of 10.
is a number that cannot be written as a sum of 3 squares.

XCIX
or IC?
Rules regarding Roman numerals often state that C cannot be preceded
by I or V, only by X. Thus, one should represent the number 'ninety-nine'
as XCIX, not as the 'shortcut' IC.
This 'problem' manifested in questions as to why 1999 was not written
simply IMM or MIM as opposed to the virtually universal MCMXCIX.

Machoumearobilengmonoolemongametsoarobilengmonoolemong means
'99' in the language of the Bassoutos tribe.

The
human brain is a busy place. We each have about 100 billion
neurons. Each of the 100 billion neurons actively communicates
with 1000 - 10,000 others.
Each of the 100 billion neurons can be 'on' or 'off', and can generate
2100,000,000,000 (or roughly 1030,000,000,000)
unique brain states. The number of the possible brain states is
greater than the number of particles in the universe!

"A Hundred Thousand
Billion Poems" (French: Cent mille milliards de
poèmes) is a set of 10 sonnets by the French writer
Raymond Queneau that can be combined into 1014 different
poems.

Only
one out of one hundredcacao
flowers will become fertilized and grow into a cocoa pod!

Indo-european
languages are classified under two major groups: the satam and
the centum group. The terms come from
the words for the number "one hundred" in
representative languages of each group. In the 'centum' group
we find italic (centum), greek (he-katon),
celtic (cent), and germanic languages (hund-rad);
in the 'satam' group: indo-iranian (satem), baltic (simtas)
and slavic languages (suto).

The
word hundred is derived from Old Norse “hundrath” which
actually meant 'long hundred’, that is 120 and not 100.

is
the only odd prime number that divides the number of bones in
a human body.

101
and 103 form a twin prime pair.

= 22 +
52 + 52 + 72

is
the smallest number other than 1 that is impossible to score
with two darts.

If
you fold a paper in half 103 times it will be
larger than the observable Universe: 93 billion light-years!
(Theoretically). But how can a 0.1-millimeter-thick paper get
to be as thick as the Universe? The answer is quite simple: exponential
growth… The paper when folded doubles, then quadruples,
then increases in thickness by leaps and bounds, which means
eight and then 16 times and then 32 times, and so on. More information here.

is
3 hyperfactorial.
can be written as the sum of a cube and a square (a3 + b2)
in two ways (33 + 92 and 23 +
102). This is the smallest number with this property.
Every angle of a pentagon measures 108°.
There are 108 different types of heptominoes.

is
a significant number in Buddhism. It represents the number of
temptations that human beings face, which tie them to the Wheel
of samsara.
There are many religious elements related to this number - a
bell is rung 108 times to bring in the New Year in Japan.

is
the sum of the numbers 4, 8, 15, 16, 23 and 42 that frequently
appear throughout the TV series 'LOST'.

The
diameter of the sun is 108 times the diameter of the Earth.

109

is
the smallest number which is palindromic in
bases 5 and 9.
Using only the coins in circulation in the UK, you can pay the
sum of 30 pence in 109 different ways.
The interior angle of a tetrahedron is 109° 28’.
Soap bubbles can only join each other at one or two angles: 109
or 120 degrees. No other angles are possible.

is
the smallest product of 3 distinct primes of the form 3n -
1:
(3x1 - 1) x (3x2 - 1) x (3x4 - 1) = 2 x 5 x 11 = 110
is the smallest number that is the product of two different substrings.
is the side of the smallest square that can be tiled with distinct
integer-sided squares (see image below). There are 3 distinct Simple
Perfect Squared Squares with this property. Two 110's with
22 squares were discovered in 1978, one by Duijvestijn using computer
search, the second by Willcocks, who transformed Duijvestijns 110
into a different second 110, and one more 110 with 23 squares was
discovered in 1990 by Duijvestijn. It was Gambini who proved 110
is the minimal square. -- Stuart
Anderson

Zeno’s
Paradox
The Greek hero Achilles and the turtle are running a foot race.
The tortoise is given a head start of 100 meters. Achilles is catching
up quickly - but the chelonian has moved a further 10 meters during
this time. Achilles quickly pulls up to this short distance as
well. But the turtle has also crawled 1 meter further... In this
way, the turtle always advances a 10th of the distance that Achilles
catches up. Can Achilles ever pass the turtle?Answer: Yes, he can! After 111 meters (after
111 1/9 meters, to be exact), Achilles will overtake the slow-moving
animal. This paradox is based on a false assumption: it does not
take into account that an infinite progression can have a finite
sum.

is
a permutable prime.
355/113 = 3.1415929... A good approximation of .

There
are 13 consecutive divisible integers (non-primes) between 113
and 127.

A113 is
an inside joke present as an Easter egg in animated films created
by alumni of California Institute of the Arts, referring to the
classroom number used by graphic design and character animation
students at the school that was used by John Lasseter and Brad
Bird among others. Brad Bird first used it for a license plate
number in the "Family Dog" episode of Amazing Stories.
It has since appeared in other Disney movies and every Pixar
movie. -- Source Wikipedia

For
Tintinologists
113, and not 57b as depicted in the comic book 'the
Calculus Affair' by Hergé, is the number of the street
where the house of the professor Alfredo Topolino is still located
in Nyon. Actually, the current full address is: 113, route de Saint-Cergue,
1260 Nyon, Switzerland.

114

appears
in the Padovan
sequence, preceded by the terms 49, 65, 86 (it is the sum
of the first two of these).

Curiosity:
Write down any number (excluding the digit 0):
64861287124425928
Now, count up the number of even and odd digits, and the total
number of digits it contains, as follows:
12 | 5 | 17
Then, string those 3 numbers together to make a new number, and
perform the same operation on that:
12517
1 | 4 | 5
Keep iterating:
145
1 | 2 | 3
You will always arrive at 123.

123
is the symbol of the Holy Trinity: Father, Son and Holy Spirit.
The number 123 is used 3 times in the Bible.

124

is
an untouchable
number.
is the smallest number such that its first 3 multiples contain
the digit 2.

is
the 4th Mersenne
prime and an autothecal number.
= 127 = 27 - 1 = - 1 + 27 (an autothecal number is an integer n that
can be expressed using all the digits of n once - in order
from left to right - , and the main math operators)

It
takes 133 years before the same phases of the moon occur on the same
day of the year and also the same day of the week.

134

1342 -
672 = 13467=
13 +23 +
53 (sum
of cubes)
= 8C1 + 8C3 + 8C4

135

=
11 + 32 + 53
= (1 x 3 x 5)( 1 + 3 +5)

136

is
the sum of the cubes of
the digits of the sum of the cubes of
its digits:136 = 23 + 43 + 4313 + 33 + 63 = 244

137

divides
11,111,111.
137 and 139 form a twin
prime pair.
it is possible to divide a pizza into 137 tiny pieces with just
17 straight cuts.
is almost the reciprocal of the Sommerfeld's fine-structure
constant (137.035999084 ± 0.000000051).
is the maximum number of 7th powers that are needed
to sum to any arbitrarily large number.
is the largest prime factor of 123,456,787,654,321.
is the average
brain weight of a baboon (in grams).

'Untrioctium'
is an unsynthesized chemical element with atomic number 138 and
symbol Uto.

is
considered as an occult number and one of its factors (23) is
also an occult number!

139

is
the number of unlabeled topologies with
5 elements.
is a prime and a happy number (see also number 86). To find out whether
a number is 'happy' or not, you must square its digits, add them
up, and go on doing this... If you eventually get to 1, you have
a happy number. For instance:
12 + 32 + 92 = 91
92 + 12 = 82
62 + 82 = 100
12 + 02 + 02 = 1

is the
the smallest number whose 5th power is a sum of four 5th powers.
This equality found in 1966 by L. J. Lander and T. R. Parkin
disproved Euler's sum of powers conjecture stating that it always
requires at least n terms of nth power to sum
to a nth power:
1445 = 275 + 845 + 1105 +
1355

Dunbar's
number is a theoretical value with implications
in sociology and anthropology. It is actually a theoretical
cognitive limit to the number of people with whom one can
maintain stable social relationships. No precise value has
been proposed for Dunbar's number, but a commonly cited approximation
is 150.

is
a triangular number:
= 1 + 2 + 3 + 4 + 5 + ... + 15 + 16 + 17
is also an hexagonal number and and one of six known truncated
triangular numbers.
= 32 x 17
= 13 + 53 + 33(is
the smallest number that is the sum of powers of its own digits,
where the power is the same as the number of digits. The next few
numbers with this property are: 370, 371, 407, 1634...)
= 1! + 2! + 3! + 4! + 5!
= 32 + 122
= 772 - 762153 = 3 x
51
153 / 999 = 0.153153153... (cyclic
number)

On
adding the number 153 to its reverse we obtain 504, whose square
is the smallest square which can be expressed as the product
of two different numbers which are reverse of one another:
153 + 351 = 504, and 5042 = 288 x 882

is
a centered triangular number.
is the number of monotone Boolean
functions of 4 variables.
is the smallest Smith
number that is prime when reversed (661), turned upside-down
(991), and reversed-then-turned-upside-down (199).

To predict
the eclipses, the ancient Siberians used the strings of dots:
{53+33}+{30}+{23}+{16}+{7+9+1+1} which are equal to 173, and
the periods of time (24 hours) -- 86, 30, 23, 16, 7, 9, 1, 1
-- are noteworthy for the calculations.
It takes around 173 years for all the water in Lake Superior to
be changed.

174

is
a 59-gonal number
= 2 x 3 x 29
= 52 + 62 + 72 + 82

175

is
the smallest number n>1 such that n6+6
is prime.
= 11 + 72 + 53

Demisemiseptcentennial or quartoseptcentennial is
a period of 175 years.

Strangely
enough, 178 and 196 are related... In fact, 178 has a square with
the same digits as 196:
1782 = 31,684
1962 = 38,416
178 has also a cube with
the same digits as 196:
1783 = 5,639,752
1963 = 7,529,536

=
22 + 22 + 72 + 112(sum
of consecutive square primes)

179

is
a prime number.

32 +
6912 + 22932 = 1793

180

is
the total number of degrees in a triangle.

arctan1
+ arctan2 + arctan3 = 180

181

is
a strobogrammatic prime.
It can be read the same upside down or when viewed in a mirror.

1812 =
1053 - 1043(solution of x2 = y3 - z3)

182

is
a palindromic and undulating number, if written in the ternary,
the negaternary, or the nonary numeral
systems.

= 33 +
33 + 43 + 43 (sum of cubes)
= 142 - 14

183

is
a central polygonal number, n2 - n +
1, where n = 14.
is a number that cannot be written as a sum of 3 squares.
is the smallest integer n so that n concatenated
with n+1 is square: 183184 =
4282

λοπαδοτεμαχοσελαχογαλεοκρανιολειψανοδριμυποτριμματο-
σιλφιοκαραβομελιτοκατακεχυμενοκιχλεπικοσσυφοφαττο-
περιστεραλεκτρυονοπτοκεφαλλιοκιγκλοπελειολαγῳοσιραιο-
βαφητραγανοπτερύγων transliterated
as: Lopadotemachoselachogaleokranioleipsanodrimhypo
trimmatosilphioparaomelitokatakechymenokichlepi
kossyphophattoperisteralektryonoptekephalliokigklo
peleiolagoiosiraiobaphetraganopterygon is a 183-letter Greek
word coined by Aristophanes in Ἐκκλησιάζουσαι (Assemblywomen);
and it represents a fictional dish created with the ingredients
which make up its name (all kinds of dainties: sauces, fish, poultry
and other meat). It is also cited as the longest ancient Greek
word ever written.

the 183th day
of the year (July 2) is the midpoint of a non-leap year (because
there are exactly 182 days before and 182 days after).

184

is
a Kaprekar
constant in base 3.
is the largest number of regions into which a plane can be divided
by 14 circles.

is
the number of digits we need to write down all numbers from 1
to 99. The general formula to find out how many digits are needed
to write down all numbers from 1 to N is:n(N + 1) - (10n - 1)/9 [where n is
the number of digits in N]

A "prime
quadruplet" is a set of 4 primes of the form p, p+2, p+6,
and p+8. Interestingly, the product of 4 primes in
a prime quadruplet (except for 5, 7, 11, 13) always ends in 189.
For instance: 11 x 13 x 17 x 19 = 46,189

= 43 +
53 = (- 3)3 + 63(sum
of cubes)

=
24 + 25 + 26 + 27 + 28 + 29 + 30 (sum of consecutive
numbers)

190

is
is a triangular number, a hexagonal number.
is the largest number with the property that it and its distinct prime factors
are palindromic in Roman
numerals:
190 = CXC = II · V · XIX- Thoms

is
a twin prime with 191, and the only known odd primep for
which 2 is not a primitive
root of 4p2+1.

A
scalene triangle having sides 193, 194 and 195
(consecutive integers) has an area that is also an integer (16296
units2). Such triangles whose
sidelengths and area are all integers are called Heronian
triangles.

=
(93 - 73) / 2

194

is
the smallest number that can be written as the sum of 3 squares in
5 different ways.

is the
product of two distinct primes: 2 x 97

= 72 +
82 + 92(sum of consecutive
squares)

195

is
the smallest number that can be expressed as a sum of distinct
squares in 16 different ways.
is the lowest value of n such that 2nCn is
divisible by n2.

196

is
a square-full number. A number is said to be square-full if for
every prime, p, that divides it, p2 also
divides it: 196 = 22 x 72

is the
lowest Lychrel number, that is the lowest number that is not
known to reach a palindrome when
repeatedly added to its reverse. For example, let's begin with
the number 392... If we reverse its digits and add it to the
original number, we get 685. Like so:
392 + 293 = 685
Since this new number is not a palindrome, we can continue the
reverse-and-add process and it may become one:
685 + 586 = 1,271
1,271 + 1,721 = 2,992 is a palindrome.
Most of the time only a few iterations (less than 10) are needed
before the number becomes a palindrome. In the case of 196, it
is believed that it will never become a palindrome, no matter how
many iterations are performed on the number. But is 196 really
immune to producing palindromes? No one’s yet offered a conclusive
proof.

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